Abstract: For a large class of piecewise expanding maps of the interval we prove the Lasota-Yorke inequality with a constant smaller than the previously known . Consequently, the stability results of Keller-Liverani apply to this class and in particular to maps with periodic turning points. One of the applications is the stability of acim's for a class of W-shaped maps. Another application is an affirmative answer to a conjecture of Eslami-Misiurewicz regarding acim-stability of a family of unimodal maps.